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If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?

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If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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New post 27 Mar 2017, 02:54
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A
B
C
D
E

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  55% (hard)

Question Stats:

60% (01:47) correct 40% (02:12) wrong based on 243 sessions

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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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New post 29 Mar 2017, 09:10
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5
Bunuel wrote:
If x^1 + x^(−1) = 5, what is the value of x^4 + x^(−4)?

A. 527
B. 546
C. 579
D. 600
E. 625


We are given that x^1 + x^(−1) = 5, i.e., x + 1/x = 5. We need to determine the value of x^4 + x^(-4), i.e., x^4 + 1/x^4.

Let’s square both sides of the equation x + 1/x = 5.:

(x + 1/x)^2 = 5^2

x^2 + 2(x)(1/x) + 1/x^2 = 25

x^2 + 2 + 1/x^2 = 25

x^2 + 1/x^2 = 23

Now let’s square the above equation:

(x^2 + 1/x^2)^2 = 23^2

x^4 + 2(x^2)(1/x^2) + 1/x^4 = 529

x^4 + 2 + 1/x^4 = 529

x^4 + 1/x^4 = 527

Answer: A
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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New post 27 Mar 2017, 03:26
1
2
Bunuel wrote:
If x^1 + x^(−1) = 5, what is the value of x^4 + x^(−4)?

A. 527
B. 546
C. 579
D. 600
E. 625


\(x^1 + x^{(−1)} = 5\) ----------------- I


\((x+1/x)^2 = x^2 + 1/x^2 + 2\)

\(5^2 = x^2 + 1/x^2 + 2\) (by using I)

\(x^2 + 1/x^2 = 23\) --------------- II



\((x^2+1/x^2)^2 = x^4 + 1/x^4 + 2\)

\(23^2 = x^4 + 1/x^4 + 2\) (by using II)

\(x^4 + 1/x^4 = 529 - 2 = 527\)

Hence option A is correct
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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New post 13 Sep 2017, 03:05
1
Official Answer

Direct attempts to solve for x in this problem will run into quadratics that don't factor and horrible non-integers that need to be raised to fourth powers. Instead, let's focus on manipulating the equation to solve for x^4 + x^−4 directly.

Given the similar structure of the given information, it seems reasonable to begin by squaring the equation x^1+x^−1=5. Be careful, though, not to simply square each term; exponents do not distribute over addition. Instead, recognize the special quadratic. We're looking at two terms added and then squared, so this expression fits the form (a+b)^2=a^2+2ab+b^2. Thus our result will be


(x^1+x^−1^2=5^2

(x^1)^2+2(x^1)(x^−1)+(x^−1)^2=25


x^2+2+x^−2=25


x^2+x^−2=23


Now just repeat the process of squaring both sides once more:


(x^2+x^−2)^2=23^2


x^4+2(x^2)(x^−2)+x^−4=23^2


x^4+2+x^−4=23^2


x^4+x^−4=23^2−2


And it's not even really necessary to calculate 23^2 (which turns out to be 529). 23^2 must end in a 9, so 23^2−2 must end in a 7, and the answer has to be A.
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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New post 05 Jan 2018, 21:46
JeffTargetTestPrep wrote:
Bunuel wrote:
If x^1 + x^(−1) = 5, what is the value of x^4 + x^(−4)?

A. 527
B. 546
C. 579
D. 600
E. 625


We are given that x^1 + x^(−1) = 5, i.e., x + 1/x = 5. We need to determine the value of x^4 + x^(-4), i.e., x^4 + 1/x^4.

Let’s square both sides of the equation x + 1/x = 5.:

(x + 1/x)^2 = 5^2

x^2 + 2(x)(1/x) + 1/x^2 = 25

x^2 + 2 + 1/x^2 = 25

x^2 + 1/x^2 = 23

Now let’s square the above equation:

(x^2 + 1/x^2)^2 = 23^2

x^4 + 2(x^2)(1/x^2) + 1/x^4 = 529

x^4 + 2 + 1/x^4 = 529

x^4 + 1/x^4 = 527

Answer: A


why do we need to square it? In the response below your original post, it says that "it's reasonable" to do. Can you explain please? Thanks
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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New post 05 Jan 2018, 21:57
rnz wrote:
JeffTargetTestPrep wrote:
Bunuel wrote:
If x^1 + x^(−1) = 5, what is the value of x^4 + x^(−4)?

A. 527
B. 546
C. 579
D. 600
E. 625


We are given that x^1 + x^(−1) = 5, i.e., x + 1/x = 5. We need to determine the value of x^4 + x^(-4), i.e., x^4 + 1/x^4.

Let’s square both sides of the equation x + 1/x = 5.:

(x + 1/x)^2 = 5^2

x^2 + 2(x)(1/x) + 1/x^2 = 25

x^2 + 2 + 1/x^2 = 25

x^2 + 1/x^2 = 23

Now let’s square the above equation:

(x^2 + 1/x^2)^2 = 23^2

x^4 + 2(x^2)(1/x^2) + 1/x^4 = 529

x^4 + 2 + 1/x^4 = 529

x^4 + 1/x^4 = 527

Answer: A


why do we need to square it? In the response below your original post, it says that "it's reasonable" to do. Can you explain please? Thanks


Hi rnz

we are given \(x^1+x^{-1}\) and need to arrive at \(x^4+x^{-4}\). So squaring the original equation will raise it to power of \(2\) i.e. \(x^2+x^{-2}\) and on further squaring this expression we will reach our destination
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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New post 06 Jan 2018, 14:54
Can someone please explain how we arrive at (x^2+1/X^2+2)?

(x+1/x)^2 = (x^2+1/X^2+2)?
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)?  [#permalink]

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New post 06 Jan 2018, 19:20
1
mekoner wrote:
Can someone please explain how we arrive at (x^2+1/X^2+2)?

(x+1/x)^2 = (x^2+1/X^2+2)?


Hi mekoner

there is a very simple formula used here

\((a+b)^2=a^2+b^2+2ab\)

now instead of \(a\) & \(b\) use \(x\) & \(\frac{1}{x}\) here :-)
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Re: If x^1+x^(−1)=5, what is the value of x^4+x^(−4)? &nbs [#permalink] 06 Jan 2018, 19:20
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